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On the Smoothness of Value Functions

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  • Strulovici, Bruno
  • Szydlowski, Martin

Abstract

We prove that under standard Lipschitz and growth conditions, the value function of all optimal control problems for one-dimensional diffusions is twice continuously differentiable, as long as the control space is compact and the volatility is uniformly bounded below, away from zero. Under similar conditions, the value function of any optimal stopping problem is continuously differentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control, which is also shown to be Markov. These conditions are based on the theory of monotone comparative statics.

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File URL: http://mpra.ub.uni-muenchen.de/36326/
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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 36326.

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Date of creation: 07 Jan 2012
Date of revision: 31 Jan 2012
Handle: RePEc:pra:mprapa:36326

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Related research

Keywords: Super Contact; Smooth Pasting; HJB Equation; Optimal Control; Markov Control; Comparative Statics; Supermodularity; Single-Crossing; Interval Dominance Order;

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References

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  1. Milgrom, P. & Shannon, C., 1991. "Monotone Comparative Statics," Papers 11, Stanford - Institute for Thoretical Economics.
  2. Juan Pablo Rincon-Zapatero & Manuel S. Santos, 2007. "Differentiability of the value function without interiority assumptions," Economics Working Papers we071405, Universidad Carlos III, Departamento de Economía.
  3. Godfrey Keller & Sven Rady & Martin Cripps, 2005. "Strategic Experimentation with Exponential Bandits," Econometrica, Econometric Society, vol. 73(1), pages 39-68, 01.
  4. John K.-H. Quah & Bruno Strulovici, 2007. "Comparative Statics, Informativeness, and the Interval Dominance Order," Economics Papers 2007-W04, Economics Group, Nuffield College, University of Oxford.
  5. Juan Pablo Rincón-Zapatero & Manuel S. Santos, 2010. "Differentiability of the value function in continuous-time economic models," Economics Working Papers we1022, Universidad Carlos III, Departamento de Economía.
  6. John K.-H. Quah & Bruno Strulovici, 2013. "Discounting, Values, and Decisions," Journal of Political Economy, University of Chicago Press, vol. 121(5), pages 896 - 939.
  7. John K.‐H. Quah & Bruno Strulovici, 2012. "Aggregating the Single Crossing Property," Econometrica, Econometric Society, vol. 80(5), pages 2333-2348, 09.
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Cited by:
  1. Keller, Godfrey & Rady, Sven, 2012. "Breakdowns," Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 396, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.
  2. Martin Szydlowski, 2012. "Ambiguity in Dynamic Contracts," Discussion Papers 1543, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

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